玛欣凯维奇函数与泊松核的一个新微分性质

On the Functions of Marcinkiewicz and Some New Differential Properties of Poisson Kernel

将Stein[On the functions of Littlewood-Paley,Lusin,and Marcinkiewicz,Trans.Amer.Math.Soc.,1958,88:430-466]中的玛欣凯维奇函数的逆向不等式推广到一般情形.主要结果是对于n-维欧几里得空间k-阶球面调和函数空间的任意一基底,得到玛欣凯维奇函数的一般性的逆向不等式,即存在不依赖于函数f正常数C_p,使得||f||_p≤C_pΣ_(j=1)~N=1||μ_j(f)||_p,其中{μ_j(f)}_(j=1)~N是f的由这些球面调和函数生成的玛欣凯维奇函数.此外,对于任意的n-变元的k-阶调和多项式Q(x)以及泊松核P_t(x),有Q(D)P_t(x)=C_n k(tQ(x))/((|x|)~2+t^2^(n+2k+1)/2).

We generalize the inverse inequality of Marcinkiewicz function in Stein[On the functions of Littlewood-Paley,Lusin,and Marcinkiewicz,Trans.Amer.Math.Soc,1958,88：430-466]to general cases.The main result of this paper is that,for any basis of spherical harmonic functions of order k in n-dimensional Euclidean space,we obtain the general inverse inequalities for Marcinkiewicz function,i.e.there exist a constant C_p does not depend on / such that ｜｜f｜｜_p≤C_p Σ_（j=1）~N=1 ｜｜μ_j（f）｜｜_p,where{μ_j（f）}_（j=1）~N are the Marcinkiewicz functions of / generated by these spherical harmonic functions.Moreover,for any n-variable homogeneous harmonic polynomial Q（x） of order k,and the Poisson kernel P_t（x）,we have Q（D）P_t（x） = C_n k（tQ（x））/（（｜x｜）~2＋t^2^（n＋2k＋1）/2）.